Diversity vs Degrees of Freedom in Gaussian Fading Channels

TL;DR

This paper redefines degrees of freedom and diversity for Gaussian fading channels using geometric and information-theoretic approaches, revealing different gauge classes and a cross-gauge capacity-diversity tradeoff in noncoherent fast fading.

Contribution

It introduces geometric definitions of DOF and diversity that are gauge-invariant and develops a cross-gauge tradeoff analysis for noncoherent fading channels.

Findings

Different gauge classes for capacity and diversity growth rates.

A cross-gauge tradeoff showing capacity on log-log scale and diversity on log scale.

Extension of the approach to coherent and block fading channels.

Abstract

The standard definitions of degrees of freedom (DOF) and diversity both normalize by $\log\rho$. When this ruler is wrong, both measurements give zero or become undefined, yet intuitively DOF and diversity ought to be channel properties, not artifacts of a normalization choice. We formalize this for Gaussian fading channels. For fixed-$H$ MIMO, DOF and diversity are both ranks of the bilinear map~$HX$ with different variables free: $\varepsilon$-covering the image of~$X\!\mapsto\!HX$ gives DOF on the $\log\rho$ gauge; expanding across all dimensions of the fading map gives diversity on the linear~$\rho$ gauge. Covering produces logs; expansion produces linear growth; so in every model studied here the two gauges differ. These geometric definitions do not yield tradeoff curves. We bridge the gap with Bhattacharyya packing, obtaining gauge-DOF and B-diversity as workable proxies -- finite and informative on every gauge, including those where the classical diversity order is zero. Three gauge classes emerge: $\log\rho$, $\log\log\rho$, and $(\log\rho)^\beta$, $\beta\in(0,1)$. The main result is a cross-gauge tradeoff for noncoherent fast fading: capacity lives on $\log\log\rho$, but B-diversity lives on $\log\rho$, exponentially larger, with matching upper and lower bounds. For coherent MIMO, block fading, and irregular-spectrum channels, the same approach recovers or extends known scaling laws.